HOW PROJECTIONS AFFECT THE DIMENSION SPECTRUM OF FRACTAL MEASURES

We introduce a new potential-theoretic definition of the dimension spe ctrum D-q of a probability measure for q > I and explain its relation to prior definitions. We apply this definition to prove that if 1 < q less than or equal to 2 and mu is a Borel probability measure with com pact support in R-n, then under almost every linear transformation fro m R-n to R-m, the q-dimension of the image of mu is min(m, D-q(mu)); i n particular, the q-dimension of mu is preserved provided m greater th an or equal to D-q(mu). We also present results on the preservation of information dimension D-I and pointwise dimension. Finally, for 0 les s than or equal to q < 1 and q > 2 we give examples for which D-q is n ot preserved by any linear transformation into R-m. All results for ty pical linear transformations are also proved for typical (in the sense of prevalence) continuously differentiable functions.